Trying to extend an edge along a curve
Overview:

The thing you are struggling here is a Pivot center for rotating/scaling.

In your situation as you stated that model isn't yours we will need to manually place 3D Cursor at the desired spot and then switch to Pivot center: 3D Cursor and we are good to go. Normally when you are creating such shape it is (mostly) in the center of the scene so the first steps wouldn't be necessary.

Steps:




Grab two vertices on the inside of the mesh like in the gif.
Press F to make an edge and connect them.
Press W > Subdivide to create vertex in the middle of created edge.
Turn on Vertex Snapping.
Press G > Y and hover mouse over the bottom most Vertex. Repeat with X Axis.
Shift+S > Cursor to Selected to place 3D Cursor where we've moved our vertex.
Delete this vertex, it's no use now.
Select Pivot center to 3D cursor.
Select inner/outer edge loop and Extrude/Scale or both.
Optionally go to Object Mode and set Origin Point to the 3D Cursor, it should help later on. Do this by pressing Ctrl+Alt+Shift+C.